Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $x \neq 0$. $t = \dfrac{x^2 - 20x + 100}{x^2 - 9x} \times \dfrac{9x - 81}{x - 10} $
Solution: First factor the quadratic. $t = \dfrac{(x - 10)(x - 10)}{x^2 - 9x} \times \dfrac{9x - 81}{x - 10} $ Then factor out any other terms. $t = \dfrac{(x - 10)(x - 10)}{x(x - 9)} \times \dfrac{9(x - 9)}{x - 10} $ Then multiply the two numerators and multiply the two denominators. $t = \dfrac{ (x - 10)(x - 10) \times 9(x - 9) } { x(x - 9) \times (x - 10) } $ $t = \dfrac{ 9(x - 10)(x - 10)(x - 9)}{ x(x - 9)(x - 10)} $ Notice that $(x - 9)$ and $(x - 10)$ appear in both the numerator and denominator so we can cancel them. $t = \dfrac{ 9\cancel{(x - 10)}(x - 10)(x - 9)}{ x(x - 9)\cancel{(x - 10)}} $ We are dividing by $x - 10$ , so $x - 10 \neq 0$ Therefore, $x \neq 10$ $t = \dfrac{ 9\cancel{(x - 10)}(x - 10)\cancel{(x - 9)}}{ x\cancel{(x - 9)}\cancel{(x - 10)}} $ We are dividing by $x - 9$ , so $x - 9 \neq 0$ Therefore, $x \neq 9$ $t = \dfrac{9(x - 10)}{x} ; \space x \neq 10 ; \space x \neq 9 $